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March 2018 SPDE limit of the global fluctuations in rank-based models
Praveen Kolli, Mykhaylo Shkolnikov
Ann. Probab. 46(2): 1042-1069 (March 2018). DOI: 10.1214/17-AOP1200


We consider systems of diffusion processes (“particles”) interacting through their ranks (also referred to as “rank-based models” in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course of the proof, we also derive quantitative propagation of chaos estimates for the particle system.


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Praveen Kolli. Mykhaylo Shkolnikov. "SPDE limit of the global fluctuations in rank-based models." Ann. Probab. 46 (2) 1042 - 1069, March 2018.


Received: 1 August 2016; Revised: 1 May 2017; Published: March 2018
First available in Project Euclid: 9 March 2018

zbMATH: 06864079
MathSciNet: MR3773380
Digital Object Identifier: 10.1214/17-AOP1200

Primary: 60H15 , 82C22 , 91G80

Keywords: central limit theorems , fluctuations in interacting particle systems , Gaussian random fields , large equity markets , mean field interaction , porous medium equation , quantitative propagation of chaos estimates , rank-based models , Stochastic partial differential equations , Stochastic Portfolio Theory , Wasserstein distances

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 2 • March 2018
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